Finite Group Databasebeta

← #7500218

Subgroups — #7500218

Order 512.

1455 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59822C211yesmaximal, characteristic
2#39635C211yesmaximal, characteristic, complemented
3#32600C214yesmaximal, complemented
4#32164C214yesmaximal, complemented
5#32080C214yesmaximal, complemented
6#30076C212yesmaximal, complemented
7#29366C212yesmaximal, complemented
8#27488C214yesmaximal, complemented
9#27360C214yesmaximal, complemented
10#26318C214yesmaximal, complemented
11#23363C211yesmaximal, characteristic, complemented
12C2×#555—21no—
13#2859C2211yescharacteristic
14#2802—24no—
15#2733—22no—
16#2647—21no—
17#2582C2212yes—
18#2548C2214yes—
19#2541C2214yes—
20#2537—21no—
21#2511C2214yes—
22#2507C2214yes—
23#2453—22no—
24#2414—24no—
25#2354C2214yes—
26Q8×C4⋊C4C2212yes—
27#2327—21no—
28C4×C4⋊Q8C2211yescharacteristic
29C4×#127C2212yes—
30#2295C2211yescharacteristic
31#2099C2212yes—
32#2096C2212yes—
33#2094C2214yes—
34#2094C2214yes—
35#2084C2214yes—
36#2071C2214yes—
37#2071C2214yes—
38#2065C2214yes—
39#2055C2214yes—
40#2041C2214yes—
41#2041C2212yes—
42#2041C2214yes—
43#2014C2214yes—
44#2014C2212yes—
45#1984C2214yes—
46#1982C2214yes—
47#1968C2214yes—
48#1957C2214yes—
49#1948C2214yes—
50#1934C2214yes—
51#1927C2214yes—
52#1926C2214yes—
53#1926C2214yes—
54#1926C2214yes—
55#1920C2214yes—
56#1916C2214yes—
57#1916C2214yes—
58#1880C2214yes—
59#1877C2214yes—
60#1864C2214yes—
61#1864C2214yes—
62#1864C2214yes—
63#1839C2212yes—
64C4×#106C2214yes—
65C22×SD16C2314yes—
66#555—48no—
67C2×C4⋊Q8D411yescharacteristic
68C2×#127C2312yes—
69C2×#127C2314yes—
70C2×#125—21no—
71C2×#124—24no—
72C2×#121—24no—
73C2×#121—24no—
74C2×C4×Q8—22no—
75C2×C4×Q8C2311yescharacteristic
76C2×C4×Q8C2312yes—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4C2314yes—
80C2×#117—24no—
81C2×#106C2314yes—
82C2×#106C2314yes—
83C2×#106C2314yes—
84C2×C4⋊C8C2314yes—
85C2×#103C2314yes—
86C2×#102C2314yes—
87C2×#102C2314yes—
88C2×#102C2314yes—
89C2×#102C2314yes—
90C2×#102C2314yes—
91C2×#102C2314yes—
92C2×#102C2314yes—
93C2×C4×C8C2314yes—
94#400—24no—
95#400—24no—
96#398—24no—
97#398—24no—
98#398—24no—
99#394—24no—
100#394—24no—
101#392—24no—
102#392—24no—
103#391—22no—
104#391D412yes—
105#391C2314yes—
106#391C2312yes—
107#390—24no—
108#390—22no—
109#390—22no—
110#390C2314yes—
111#390C2314yes—
112#390C2314yes—
113#389—22no—
114#389—22no—
115#389—24no—
116#389—22no—
117#389—24no—
118#389D414yes—
119#389C2314yes—
120#389C2314yes—
121#389C2312yes—
122#388—24no—
123#388—24no—
124#387—24no—
125#387—24no—
126#387—22no—
127#387—24no—
128#384D411yescharacteristic
129#384C2312yes—
130#384C2312yes—
131#384C2314yes—
132#382—21no—
133#380—24no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2314yes—
139#376—21no—
140C43C2311yescharacteristic
141#340C2314yes—
142#340C2314yes—
143#340C2314yes—
144#340C2314yes—
145#336C2314yes—
146#336C2314yes—
147#336C2314yes—
148#336C2314yes—
149#336C2312yes—
150#336C2312yes—
151C22×Q8C2×D412yes—
152C22×Q8C2411yescharacteristic
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4C2414yes—
158C23×C4—24no—
159C2×SD16—44no—
160C2×SD16—44no—
161C2×SD16—44no—
162C22×C8C2414yes—
163C22×C8C2414yes—
164C4⋊Q8SD1618yescomplemented
165#127—44no—
166#127—44no—
167#127—44no—
168#125—82no—
169#124—416no—
170#121—416no—
171#121—88no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×Q8—84no—
176C4×D4—44no—
177C4×D4—44no—
178C4×D4—48no—
179C4×D4—84no—
180#117—416no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D414yes—
193C2×C4⋊C4C2×D414yes—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2414yes—
196C2×C4⋊C4C2411yescharacteristic
197C2×C4⋊C4C2412yes—
198C2×C4⋊C4C2414yes—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C42—24no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—24no—
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2412yes—
214C2×C42C2411yescharacteristic
215C2×C42C2414yes—
216#106—44no—
217#106—44no—
218#106—44no—
219C4⋊C8—44no—
220#103—44no—
221#102—44no—
222#102—44no—
223#102—44no—
224#102—44no—
225#102—44no—
226#102—44no—
227#102—44no—
228C4×C8—44no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—24no—
233#95—21no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95C2×D414yes—
240#95C2×D414yes—
241C24—24no—
242C2×Q8—24no—
243C2×Q8—22no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8C2×SD1618yes—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—48no—
258C2×D4—82no—
259C2×D4—82no—
260C2×D4—84no—
261C2×D4—84no—
262C2×D4—84no—
263C2×D4—84no—
264C2×D4—84no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—21no—
269C22×C4—24no—
270C22×C4—48no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4C22×D414yes—
274C22×C4C22×D412yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D412yes—
277C22×C4C22×D414yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C2511yescharacteristic
280SD16—88no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—84no—
302C4⋊C4—84no—
303C4⋊C4—82no—
304C4⋊C4—84no—
305C4⋊C4—84no—
306C4⋊C4C2×SD16116yes—
307C22⋊C4—48no—
308C22⋊C4—48no—
309C22⋊C4—48no—
310C22⋊C4—48no—
311C22⋊C4—84no—
312C22⋊C4—84no—
313C22⋊C4—84no—
314C42—24no—
315C42—24no—
316C42—24no—
317C42—22no—
318C42—42no—
319C42—41no—
320C42—42no—
321C42—42no—
322C42—48no—
323C42—84no—
324C42C2×SD1614yes—
325C23—44no—
326C23—44no—
327C23—44no—
328C23—44no—
329C23—44no—
330C23—44no—
331C23—44no—
332C23C23×D411yescharacteristic
333Q8—48no—
334Q8—44no—
335D4—44no—
336D4—44no—
337D4—44no—
338D4—44no—
339D4—88no—
340D4—88no—
341D4—168no—
342D4—168no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—21no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—21no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—816no—
371C2×C4C22×SD1618yes—
372C2×C4C22×SD1612yes—
373C2×C4C22×SD1614yes—
374C2×C4#58512yes—
375C2×C4#58514yes—
376C8—44no—
377C8—44no—
378C22—84no—
379C22—84no—
380C22—84no—
381C22—84no—
382C22—84no—
383C22—84no—
384C22—84no—
385C22#348511yescharacteristic
386C22#348712yes—
387C22C23×SD1611yescharacteristic
388C22#358511yescharacteristic
389C22#358512yes—
390C4—28no—
391C4—22no—
392C4—24no—
393C4—41no—
394C4—44no—
395C4—42no—
396C4—81no—
397C4—81no—
398C4—84no—
399C4—82no—
400C4—84no—
401C4#356614yes—
402C2—164no—
403C2#5208411yescharacteristic
404C2#6169711yescharacteristic
405C2#6179711yescharacteristic
406C2#6188112yes—
407C2#6193812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59822C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #39635C2semidirect
3. #32600C2semidirect
4. #32164C2semidirect
5. #32080C2semidirect
6. #30076C2semidirect
7. #29366C2semidirect
8. #27488C2semidirect
9. #27360C2semidirect
10. #26318C2semidirect
11. #23363C2semidirect
164. C4⋊Q8SD16semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#598222
#396352
#326002
#321642
#320802
#300762
#293662
#274882
#273602
#263182
#233632